There is no largest prime number.

Established within a clearly defined formal system, or by an explicitly stated standard of proof appropriate to the domain.

A checkable derivation from stipulated axioms, or an equivalent domain-specific demonstration. Empirical theories are not labelled proven merely because evidence is abundant.

The proof is elementary and independently checkable; confidence is not a probability.

This is an epistemic verdict, not a claim that the thesis has been proven unless its status is “Proven”. “Valid” in logic is a separate property of arguments, assessed in the arguments section.